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Submitted on 1 Jan 1978
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FINITE SUSCEPTIBILITY PHASE IN THE
DISORDERED TWO DIMENSIONAL X-Y MODEL
J. José
To cite this version:
J. José.
FINITE SUSCEPTIBILITY PHASE IN THE DISORDERED TWO
JOURNAL DE PHYSIQUE Colloque C6, supplPment au no 8, Tome 39, aolit 1978, page C6-749
FINITE SUSCEPTIBILITY PHASE IN THE DISORDERED TWO DIMENSIONAL X-Y MODEL
The James Franck I n s t i t u t e , The University of Chicago, Chicago, I L 60637
-
USAR6sumb.- A partir du rbsultat obtenu pour les fonctions de correlation de spin du modsle X-y dbsor- donn6 B d e w dimensions, nous suggbrons l'existence de phase paramagnltique, apparemment pour toutes les tempbratures, pour la densit6 X = 112 des interactions antiferromagnbtiques.
Abstract.- From a calculation of the spin-spin correlation functions of the disordered 2-d X-y model we suggest the existence of a paramagnetic phase for a density X = 112 of antiferromagnetic bonds, apparently for all temperatures.
Our analysis is based on the concept of frus- tration as applied to the planar model. The idea of frustration was introduced by Toulouse in his attempt
to single out the most important property of the SG phase /l/. The ke? point in this analysis is the re- cognition that a frustrated plaquette is equivalent to a half integer quenched vortex. This point was stressed by Villain /2/. Before discussing the pro- perties of the frustrated planar model, a brief re.count of our present understanding of the behavior of the purely ferromagnetic X-y model will be given /3,4/. The low temperature phase is characterized by the coexistence of spin waves plus dual vortex- like excitations that appear as bound states with opposite vorticities. The potential of interaction between vortices is logarithmic and with strength proportional to T-l. So, as we increase T the vor- tices separate until we reach a critical temperature, T = a/2, for which the pair unbinds. Above Tc, there is presumably a "conducting-like" phase of vortices. The correlations between two far separa- ted spins decay algebraically from T = 0 to T = ~ / 2 , that is the correlations remain critical in thisin- terval of temperatures with an exponent
n l
which is a function of both T and the density, y, of vortices. The vortices are irrelevant variables in the sense of! the renormalization group for all T5
Tc, but are responsible for a finite value of Tc.f
James Franck Fellow 1977-79. Work supported in part by NSF Grant DMR 77-12637.
.(XI leave from Facultad de Ciencias, UNAM,
Mexico.
Consider a purely ferromagnetic state and replace one bond by an antiferromagnetic one. In terms of frustration a nearest neighbour pair of merons / S / has been added /l/. As the density, X, of negative bonds is increased, two things happen ;
first, the number of meron pairs grows and second the separation distance of the members of a given pair increases accordingly. Then, the system (for y = 0) consists of a gas of meron dipoles with an
essentially continuously varying range of sizes and corresponding dipole moments.
When we include both the vortices and merons, the picture is as follows. The vortices interact logarithmically with both the merons and other vor- tices. In fact, a vortex would not be able qualita- tively to distinguish if it is bounded to a meronor to another vortex as we raise the temperature. This should, of course, be such that the total charge neutrality condition (meron charge + vortex charge
= 0) is satisfied. One of the important points of
the calculation for the spin-spin correlation func-
tions is to monitor the behavior of the vortex- vortex (W) correlation function as the temperature is increased from zero / 4 / . Here, we want to know the behavior of the v-v correlation function inthe presence of the background potential generated by
the presence of the meron dipoles. To include all
possible interactions is a difficult task. However, the calculation can be performed by considering the following simplifying features. Denote by .?! the
separation distance between the centers of the me- rons forming a given dipole and D the corresponding distance for a vortex pair. Let s be the separation distance between the meron dipole and vortex pair.
If S >> D and S >>
R
,
then the interaction will be and carrying out the configurational average that a dipole-dipole one, which decays like l/s2. In thecase where one of the members of the meron dipole is in the boundary of the system, its partner will look like an isolated meron and the energy of inter- action with a vortex pair decays like 11s. For an isolated meron and a vortex, the energy of inter- action goes like Ens. There are, of course, interme- diate regimes with more complicated behavior. The main approximation in this paper is to keep only the logarithmic interactions.
The actual calculation of the correlation function is based on the heuristic picture given above and is quite involved. Details can be found elsewhere 161.
Here, however, we quote some of the main results of the calculation. The spin-spin correla- tion function is obtained from averaging /6/
+ + i .I~I/~F(%)U(~)- + +
g(r. r', {F)) = e g<r, r1 ; { F ( % ) (1)
with respects the configuration of meron charges. 112 F($) gives the meron charge at the dual lattice
+ -+
site R and takes values of f 112. U(R) is the poten- tial function definedinequation 4.25 of reference 4. " g s a configurational spin-spin correlation func- tion obtained from a thermal average with respect to a Hamiltonian that includes the generalized-Villain Hamiltonian for vortices introduced in reference 4 and the logarithmic coupling between vortices and merons. A perturbation expansion is developed for small densities, xf, of frustration. The result is that there is an infinite low temperature suscepti- bility phase at low temperatures as in the ferro- magnetic case but the critical temperature goes down
2
from Tc = n/2 to T c l = 5 n to lowest order in X
f' The higher order corrections to this result are difficult to carry out. However, at the special point X = 112 the averaging of (1) can be carried out. At this point the probability of having a frustrated plaquette is equal to 112. Care should be taken of the fact that merons appear in pairs. This constraint can be introduced at the level of the probability function.
-
+g(r,
t'
; {F(%))) is evaluated within acumulant expansion valid for low densities, y, of thermally excited vortices. The effect of the merons appear in the vortex-vortex configurational corre- lation function which is evaluated to lowest order in y 2
.
Substituting the calculated in equation (I)- + +
is denoted by g(r, r') the result, to leading order, is
with K = exchange integralltemperature. Equation
(2) gives the main result of this communication. It
shows that for X = 112 the correlation functions
decay exponentially and therefore the susceptibili- ty
X
shows a paramagnetic ( X - K ) behavior. The eva- luation of (2) was done at low temperatures, however, because we see no detectable effect that would in- dicate a phase transition for K < m we are led tosuggest that
H
decays exponentially at all tempera-tures. Two calculations were performed ; for
xf << 1 and for xf = 112. When xf << 1 the behavior was similar to the ferromagnetic one but with a
lower critical temperature. It should be expected that for higher densities of xf the critical tempe- rature should go down until it reaches zero at a critical value xc. From our analysis we are not able to decide if X = 112 is the special point X
C'
from which the system gaes from an infinite suscep- tibility phase at sufficiently low temperatures to the finite susceptibility phase for K > m
.
Fromthe technical point of view, however, X = 112 is a special point. More results and details of this calculation are given in reference 161.
Acknowledgements.- The author wishes to thank G.S. Grest, J. Hertz, and L.P. Kadanoff for very helpful conversations.
References
/ l / Toulouse, G. Commun. Phys.
2
(1977) 115 /2/ Villain, J., J. Phys. C. Solid.St.Phys.10
(1977) 4793
/3/ Kosterlitz, J . M . , and Thouless, D . J . , J. Phys.
C. Solid.St. Phys. 6 (1973) 1181 ; Kosterlitz,
J.H., ibid.,
1
(197z) 1046/4/ Jos6, J., Kadanoff, L., Kirkpatrick, S. and
Nelson, D . , Phys. Rev. B
16
(1974) 1217/S/ We prefer to use the term merons instead of
"half integer quenched vortex" and will leave the term vortices for the thermally excited ones.
